1. Field of the Invention
The present invention relates to an analysis apparatus, an analysis program product, and a computer readable recording medium having an analysis program recorded thereon. More particularly, the present invention relates to an analysis apparatus, an analysis program product, and a computer readable recording medium having an analysis program recorded thereon, for obtaining a current value, a voltage value, an electric field intensity, and a magnetic field intensity, using Finite Difference Time Domain (hereinafter referred to as FDTD method) for differentiating and solving Maxwell differential equation with a time domain in cooperation with a circuit analysis typified by SPICE (Simulation Program with Integrated Circuit Emphasis developed by the University of California at Berkeley).
2. Description of the Background Art
The conventional analysis system includes a plurality of information processors connected through a network for processing divided regions to be analyzed in parallel, as disclosed in Japanese Patent Laying-Open No. 2004-054642. The analysis system includes at least two information processors for analyzing the divided region to be analyzed based on FDTD method in terms of an electric field and a magnetic field, and at least two information processors for analyzing an equivalent circuit of the divided region to be analyzed by means of a circuit simulation. The information processor for analyzing an electric field and a magnetic field includes a first arithmetic circuit operating at least one of the strength of an electric field and the strength of a magnetic field at a prescribed computational time in an overlap region, a transmission circuit transmitting at least one of the strength of an electric field and the strength of a magnetic field in the overlap region at a prescribed computational time to the information processor analyzing the adjacent region, and a second arithmetic circuit operating at least one of the strength of an electric field and the strength of a magnetic field at a computational time in a region other than the overlap region of the divided regions to be analyzed, in parallel with the transmission performed by the transmission circuit.
In this analysis system, it is assumed that an element analyzed by a circuit simulator is sufficiently small in size as compared with a wavelength and exists axially parallel in one cell. The Amère law expressed by formula (1) can be rewritten to formula (2) for an FDTD cell including an element.
                                          ɛ            ⁢                                          ⅆ                E                                            ⅆ                t                                              +                      J            ⁡                          (              E              )                                      =                  ∇                      ×            H                                              (        1        )                                                      C            ⁢                                          ⅆ                V                                            ⅆ                t                                              +                      I            ⁡                          (              V              )                                      =        I                            (        2        )            
Here, V is a voltage applied to a chip, C=εA/Δz is the electrostatic capacity of the FDTD cell (A=Δx·Δy is a cell area of FDTD and Δz is a height thereof), I(V) (=AJ(E)) is current flowing in an element, and I is the entire cell current A·∇×Hn+1/2. In other words, the combination of the information processor analyzing based on FDTD method and the circuit simulator is expressed by a capacitor C, a constant-current source I, and an equivalent circuit of an element, which are connected in parallel. FIG. 9 shows the equivalent circuit thereof.
Data passing between the information processor analyzing based on FDTD method and the circuit simulator in the equivalent current source method will now be described. FIG. 10 is a data flow in the information processor analyzing based on FDTD method and the circuit simulator in time sequence.
Assume that electric field intensity En−1 at a time (n−1)Δtem and magnetic field intensity Hn−3/2 at a time (n− 3/2)Δtem are known. Magnetic field intensity Hn−1/2 at a time (n−½)Δtem can be found by substituting En−1, Hn−3/2 into formula (4) described later. However, the electric field intensity En at a time nΔtem is calculated differently between a cell including an element and a cell including no element. In a cell including no element, the electric field intensity En is calculated by substituting En−1 and Hn−1/2 in formula (3) described later. In a cell including an element, the electric field intensity En is calculated by a circuit analysis using a circuit simulator. The circuit simulator calculates electric field intensity En by performing a circuit simulation from time (n−1)Δtem to time nΔtem (the time intervals are set finely enough) where an initial voltage value is Vn−1=Ezn−1Δz and an equivalent current source value is I=A·∇×Hn−1/2. The voltage value Vn of an element at time nΔtem is converted into electric field intensity of a cell including an element, Ezn=Vn/Δz, which is passed to the information processor analyzing based on FDTD method.
In the equivalent voltage source method, similarly, an equivalent voltage source value is obtained from an electric field obtained by FDTD method, and a current value at a time at which a magnetic field is obtained by FDTD method is analyzed by the circuit simulator, converted into a magnetic field, and then passed to the information processor analyzing based on FDTD method for further analysis.
The circuit simulator will be illustrated briefly. The circuit simulator is generally used as a tool for analyzing a transient state of an electric circuit including a nonlinear element. A library that covers a great number of subcircuits including integrated circuits is provided by manufactures, software companies, universities concerned, and the like. The analysis method in the circuit simulator is as follows. First, a current/voltage value at a node of a circuit to be analyzed is given as a variable, and Modified Nodal Analysis is adopted to a netlist on which information of connection between circuit elements and parameters of circuit elements are described, whereby nonlinear simultaneous differential equations are derived. These are converted into algebraic equations using a difference in a time domain and Newton iterative method. Those algebraic equations are solved to obtain the current/voltage value of the circuit at an analysis time. The transient state of voltage/current of the circuit is then obtained by advancing the difference time in the time domain and repeating the above calculation.
In the analysis using the method as described above (a method of analyzing electric field intensity, magnetic field intensity, and a circuit transient response in an analysis region including a nonlinear circuit element by a numerical simulation in which FDTD method and a circuit simulator cooperate with each other in a time domain. The method will be referred to as “the hybrid method” hereinafter), the electric field intensity and the magnetic field intensity in FDTD method are related to the current value and the voltage value in the circuit simulator.
Furthermore, Japanese Patent Laying-Open No. 2004-054642 describes a method of processing numerical calculations in parallel for the purpose of a fast analysis in which the hybrid method is expanded to a circuit analysis system configured with a plurality of information processors over a network.
In the following, the hybrid method in Japanese Patent Laying-Open No. 2004-054642 will be described with reference to FIG. 11. It is noted that FIG. 11 illustrates distribution of the hybrid-method processing for each information processor.
First, an analysis target including a circuit element is divided into a plurality of regions depending on the operation ability of a computer (1) and a computer (2) that conduct an analysis based on FDTD method, and is then allocated to the computers. In this example, the distribution to computer (1) and computer (2) is in the proportion of 5:4. In addition, an overlap region 130 is provided at the boundary of each region. An absorbing boundary region 132 is provided at the outer periphery of each region.
In each computer that conducts an electromagnetic field analysis, after the magnetic field intensity in each region is computed, the magnetic field intensity in overlap region 130 and then the electric field intensity in overlap region 130 are preferentially computed. The electric field intensity and the magnetic field intensity in overlap region 130 are transferred to the computer that analyzes the adjacent region based on FDTD method. Thereafter, the computations of the electric field intensity and then the magnetic field intensity in region (1) and region (2) are processed in parallel by computer (1) and computer (2). After the computation of the magnetic field intensity, the electric field intensity and the magnetic field intensity in overlap region 130 are received from the computer that has analyzed the adjacent region, so that the magnetic field intensity and then the electric field intensity at the next time are computed. Thereafter, the electromagnetic field analysis is performed repeatedly until a desired time.
Here, the circuit analysis for use in the hybrid method is also allocated to a computer (3) and a computer (4) for each region. The circuit analysis is performed in parallel with FDTD method in the following order. First, after the magnetic field intensity of overlap region 130 is computed, a constant-current value in the hybrid method is calculated from the magnetic field intensity on the periphery of an element (A) 134 and an element (B) 136. Then, the voltage values of element (A) 134 and element (B) 136 are calculated by solving circuit equations of respective regions in parallel. Then, the voltage values of element (A) 134 and element (B) 136 are converted into the electric field intensity (the electric field intensity in a cell including an element). Then, the electric field intensity is transferred to computer (1) and computer (2) that perform FDTD method. Thereafter, the operations are repeatedly performed in parallel by the computers as described above.
In accordance with this invention, the hybrid method is employed in the analysis system (configured with a plurality of information processors over a network), thereby achieving a higher speed.
An information processor according to Japanese Patent Laying-Open No. 2003-223426 analyzes an electric field and a magnetic field by FDTD method together with a plurality of other information processors, in which a region to be analyzed is divided. The information processor includes: a first arithmetic unit operating at least one of the strength of an electric field and the strength of a magnetic field at a prescribed computational time in an overlap region of divided regions in which at least one of a plurality of other information processors individually analyzes an electric field or a magnetic field; a transmission controller controlling transmission of at least one of the strength of a magnetic field and the strength of an electric field in the overlap region at a computational time as operated by the first arithmetic unit to the other information processor analyzing the overlap region; and a second arithmetic unit operating at least one of the strength of an electric field and the strength of a magnetic field at a computational time in a region of the divided regions except the overlap region in parallel with the transmission controlled by the transmission controller.
In accordance with this invention, an electric field and a magnetic field can be analyzed by FDTD method more quickly using a plurality of information processors, without degradation in precision.
Toru Uno, “Finite Difference Time Domain Method for Electromagnetic Field and Antenna Analysis”, the first edition, Corona Publishing, Mar. 20, 1998, pp. 2–10 (referred to as “Reference 1” hereinafter) discloses the hybrid method of simulating an electromagnetic wave radiating from electronic equipment using FDTD method. In accordance with this invention, an electromagnetic wave can be simulated more easily using a simple algorithm.
FDTD method will be described briefly with reference to FIG. 12. In FDTD method, when a structure such as a printed circuit board or a cavity of electronic equipment and the surrounding space are assumed as an analysis region, the analysis region is divided into minute cuboids called cells shown in FIG. 12. Here, each cell is given with magnetic permeability, permittivity, and conductivity according to the substance that forms the cell. The length of each side in the x, y, z direction is set as Δx, Δy, Δz, respectively.
Then, each component (Ex, Ey, Ez) of x, y, z of one of the electric field intensity E and the magnetic field intensity H as vector quantity (in the following description, the electric field intensity) is arranged on each side of the lattice of the cell. Each component (Hx, Hy, Hz) of x, y, z of the other (in the following description, the magnetic field intensity) is arranged vertically to the lattice face in the middle of the lattice face of the cell.
The following two formulas are obtained by using a central difference for time and space for the Maxwell differential equation.
                              E          n                =                                                            1                -                                                      σ                    ⁢                                                                                  ⁢                    Δ                    ⁢                                                                                  ⁢                    t                                                        2                    ⁢                                                                                  ⁢                    ɛ                                                                              1                +                                                      σ                    ⁢                                                                                  ⁢                    Δ                    ⁢                                                                                  ⁢                    t                                                        2                    ⁢                                                                                  ⁢                    ɛ                                                                        ⁢                          E                              n                -                1                                              +                                                                      Δ                  ⁢                                                                          ⁢                  t                                ɛ                                            1                +                                                      σ                    ⁢                                                                                  ⁢                    Δ                    ⁢                                                                                  ⁢                    t                                                        2                    ⁢                                                                                  ⁢                    ɛ                                                                        ⁢                          ∇                              ×                                  E                                      n                    -                                          1                      2                                                                                                                              (        3        )                                          E                      n            +                          1              2                                      =                              E                          n              -                              1                2                                              -                                                    Δ                ⁢                                                                  ⁢                t                            μ                        ⁢                          ∇                              ×                                  E                  n                                                                                        (        4        )            
Here, σ, ε, μ indicate conductivity, permittivity, and magnetic permeability, respectively.
FIG. 13 shows the order in which the electric field intensity and the magnetic field intensity are calculated corresponding to the passage of time in FDTD method. Assume that a time interval is Δtem, and the electric field intensity En−1 at a time (n−1)Δtem and the magnetic field intensity Hn−1/2 at a time (n−½)Δtem are known. The electric field intensity En at a time nΔtem is calculated by substituting En−1, Hn−1/2 into formula (3). Subsequently, the magnetic field intensity Hn+1/2 is calculated by substituting En, Hn−1/2 into formula (4) at a time (n+½)Δtem. In this manner, the electric field intensity E and the magnetic field intensity H are calculated alternately in time in FDTD method.
Here, time interval Δtem in FDTD method needs to satisfy the Courant stability condition shown in formula (5) for the size of a cell.
                              Δ          ⁢                                          ⁢                      t            em                          ≤                  1                      c            ⁢                                                                                (                                          1                                              Δ                        ⁢                                                                                                  ⁢                        x                                                              )                                    2                                +                                                      (                                          1                                              Δ                        ⁢                                                                                                  ⁢                        y                                                              )                                    2                                +                                                      (                                          1                                              Δ                        ⁢                                                                                                  ⁢                        z                                                              )                                    2                                                                                        (        5        )            
Here, c is the speed of light. It is generally known that when formula (5) is not satisfied for the time interval, the calculated value diverges.
Furthermore, when an open region is handled, reflection takes place at an external wall of an analysis region, causing an error. This problem is caused by that FDTD method is an analysis approach for a closed region. For this problem, a virtual boundary called an absorbing boundary needs to be provided in order to prevent reflection at an external wall of an analysis region. Reference 1 also introduces various absorbing boundary conditions that have been suggested so far.
In the invention disclosed in the aforementioned Japanese Patent Laying-Open No. 2004-054642, however, when the number of elements handled in the circuit analysis increases, the analysis target including a large number of circuit elements cannot be analyzed at high speed. This is because the time required for the circuit analysis increases. The invention disclosed in Japanese Patent Laying-Open No. 2003-223426 has also the similar problem. This problem will be described more specifically. The algebraic equation as converted in the course of the analysis using the circuit simulator is computed using a matrix operation. The order of this matrix becomes equal to the number of nodes of the circuit. Generally, when linear simultaneous equations with n unknowns are solved using a solution such as Gaussian elimination, the complexity is about n3/3+O(n2). As the scale of a circuit to be analyzed increases and the number of nodes increases, the complexity dramatically increases. This is the cause of an increase in analysis time. The increased analysis time also increases the computational cost.
In the invention disclosed in Reference 1, the computation time is almost determined by the operation speed of CPU (Central Processing Unit) and memory, irrespective of the number of elements, and high-speed analysis is not achieved.